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A NOTE ON LIFE TABLE AND MULTIPLE-DECREMENT TABLE FUNCTIONSAuthor(s): W. F. ScottSource: Journal of the Institute of Actuaries (1886-1994), Vol. 117, No. 3 (DECEMBER 1990), pp.671-675Published by: on behalf of theCambridge University Press Institute and Faculty of Actuaries
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A
NOTE ON
LIFE
TABLE AND
MULTIPLE-DECREMENT
TABLE
FUNCTIONS
By
W. F.
Scott,
M.A.,
Ph.D.,
F.F.A.
ABSTRACT
This note onsiders
he
mortality
able
functions,
nd
shows hat heusual formulae old under ess
restrictive
ssumptions
han those
usually
made. The foundations
f the
theory
of
multiple-
decrementables are also considered,n thecontext fprobabilityheory.
KEYWORDS
Life
Tables;
Decrement ables.
1. INTRODUCTION
The
properties
f the
mortality
able functions re
usually
derivedunder the
assumption
hat
i
exists nd is
continuous
see
Neill,
Section
1
6).
We
shall how
that
heusual
formulae oldunder
essrestrictive
ssumptions,
iz. that
x
nd
fix
exist nd are continuous. hemathematical asis for his pproach s similar o
that sed n
connection
ith he
force f
nterest
n
McCutcheon&
Scott,
ection
2.4.
We
also consider
he
foundations f
the
heory
f
multiple-decrement
ables,
whichwe
attempt
o
place
n
the
ontext f
probability
heory.
n
particular,
e
prove
the
identity
f
the forces'
see
Hooker
&
Longley-Cook,
ection
20.7).
2. SOME
MATHEMATICAL
RESULTS
Theorem .1
Let /=
[A,B)
or
-
oo,
2?),
where
B
may
be oo
Suppose that/(x)
s continuous
on
/and that ts
right-hand
erivative,/+x),
is zero on /.
Then/(;c)
s constant.
Proof (Adapted
from
Hobson,
p.
365)
Let us consider he
ase when
/=
(
-
oo,
/?);
he
ase when
/=
[A,B)
s
similarly
dealt with. f
the
onclusion f the
theorem e
false,
heremustbe a
and b
in
/,
with
(x,k)=f(x)-f(a)-k(x-a)
which s
continuous n /.
Note that
t>{a,k)
0. Let k
be
positive
ut o small
hat
(/>(b,k)2q>0.Let M =
{x:
a
7/23/2019 A Note on Life Table and Multiple-Decrement Table Functions
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672
A Note on
Life
Table
and
Multiple-
ecrement
a(,&)
q.
Now let
xn-+
- Since
q
for
ll
n,
/>(,
)>q,
and so
(j)(,k)
q.
Also:
/i
x
X,?
-
But
this ontradicts
he fact hat
p'+(x,k)=
-k
7/23/2019 A Note on Life Table and Multiple-Decrement Table Functions
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Table
Functions
673
We also define:
x
=
1-tPx
=
Pr{a
life
ged
x
does
not survive o
age
x
+
t}.
(3.4)
(If
t=
1,
we
may
write
qx
qx
and
tpx=px.)
We assume
that,
for
x>a:
Pr{a
life
ged
x dies
within ime
A}
ix
=
hm
//-0+
A
= lim ^ (3.5)
/i-o+ A
exists,
and is
a continuous function n
[a,oo).
We also assume
that,
for
OL
7/23/2019 A Note on Life Table and Multiple-Decrement Table Functions
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674
A Note on
Life
Table and
Multiple-
ecrement
Formula 3.8
now follows
rom
his
nd formula .7.
Q.E.D.
It now follows
asily
on substituting
=
'nx+sds)
that:
o
hqxJ'tPxHx
dt.
(3.9)
0
Now let
x and t be
fixed,
with
t25,
say,
s(x)
is continuous nd
that:
Pr{a
life
ged
x
dies
within ime
h}
=
Bcx
h
+
o(h).
This
implies
hat:
i.e.
Gompertz'
aw
holds
for
x>25,
and
xx
s
clearly
ontinuous.
f
we also
suppose
thatformula .6
holds,
Theorem3.1 showsthat,forx>25 and />0:
,p,
exPr-fr*+'&l
=
exp[-5rv(c'-l)/logc].
4.
THE MULTIPLE-DECREMENT
TABLE
Let usconsider nly womodesofdecrement,andy, ndsupposethatwe are
mainly
nterested
n
mode
.
(If
there
re
more han
wo
modes,
ll
except
mode
may
be
combined.)
t s
supposed
hat
ach mode
ofdecrement
as the
life able'
functions
x?, px,
tc.,
as
in
Section
3.
Lives are
considered
to continue
n
existence
ith
espect
o a
given
mode after
xit
by
theother
mode ofdecrement.
To avoid
philosophical
roblems
when
one of the
modes of
decrement
s
death,
one
may,
for
xample,
ake
mode
as exit
by marriage
nd
mode
y
as
exit
by
withdrawal
rom
ervice
mong
thebachelor
mployees
f
a
large
organisation,
mortality
eing gnored.
Let
Tu T2
denote
he imes
o exit
by
modes
and
y respectively
or
life
ged
x.
It follows
by
formula
3.9
that
Tu T2
have
probability ensity
functions
txP%Px
u
(t'
>
0)
and
t2pl'i'
t2
t2
>
0) respectively.
e assumethatTx nd T2
are
independent,
nd
define:
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Table
Functions
675
t(ap)x
the
probability
hat life
ged
jc,
subject
o both
modes of
decrement,
survives o
age
x+t
=
tp.'tpi (4.1)
by
the
ndependence
f
T'
and
T2.
If
/
1
we
may
omit
t.)
We also
define:
t(aq)x
the
probability
hat life
ged
x,
subject
o both modes of
decrement,
will
xit
by
mode
before
ge
x+
1,
xit
by
mode
y
not
having reviously
occurred
=
Pr{r